The Heaviside function H is defined by H ( t ) = { 0 if t < 0 1 if t ≥ 0 It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage V(t ) in a circuit if the switch is turned on at time t=0 and 120 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (c) Sketch the graph of the voltage V(t ) in a circuit if the switch is turned on at time t=5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H ( t ). (Note that starting at t = 5 corresponds to a translation.)
The Heaviside function H is defined by H ( t ) = { 0 if t < 0 1 if t ≥ 0 It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage V(t ) in a circuit if the switch is turned on at time t=0 and 120 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (c) Sketch the graph of the voltage V(t ) in a circuit if the switch is turned on at time t=5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H ( t ). (Note that starting at t = 5 corresponds to a translation.)
It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on.
(a) Sketch the graph of the Heaviside function.
(b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t=0 and 120 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t).
(c) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t=5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (Note that starting at t = 5 corresponds to a translation.)
As the speed of a train increases, the amount of power needed to maintain that speed increases, and the rate
of power increase also increases. Let P = f(v) be the power, in megawatts, needed for the train to maintain a
speed of v kilometers per hour.
(a) Sketch a possible graph of f.
(b) For each of the functions f, f', and f", decide whether the function is positive or negative, and explain
your answers.
(c) What requires a greater increase in power: increasing the train's speed from 100 to 150 kilometers per
hour, or increasing the train's speed from 200 to 250 kilometers per hour? Explain, based on your
answer to part (b).
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